Finite Group Databasebeta

← #16935497

Subgroups — #16935497

Order 4352.

670 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763157C211yesmaximal, characteristic
2#14533783C211yesmaximal, characteristic
3#14529737C211yesmaximal, characteristic, complemented
4C2×#11762552C211yesmaximal, characteristic
5C2×Q8⋊#3622C211yesmaximal, characteristic, complemented
6#14526623C211yesmaximal, characteristic, complemented
7C2×C17×#462C211yesmaximal, characteristic, complemented
8C2×#10587560C2211yescharacteristic
9C2×#10587633—21no—
10#11763157—22no—
11#11763157—22no—
12C2×C4⋊#3622C2211yescharacteristic
13C2×#10587480—22no—
14C2×C4×#3622C2211yescharacteristic
15#11762726—22no—
16#11762711—22no—
17#11762710—21no—
18C4×C4⋊Dic17C2211yescharacteristic
19#11762571—22no—
20#11762552—22no—
21Q8⋊#3622—22no—
22C22×Q16×C17—21no—
23C2×C17×C4⋊Q8C2211yescharacteristic
24C2×C4×Q8×C17C2211yescharacteristic
25C17×#462—22no—
26C17×#462—22no—
27C2×C17×C4⋊C8C2211yescharacteristic
28C2×C17×#103—21no—
29C22×Q8×C17—21no—
30C22×Q8×C17D411yescharacteristic
31C22×D68—21no—
32C22×D68—21no—
33C22×D68D412yes—
34C2×Q16×C17—24no—
35C2×Q16×C17—42no—
36C2×Q16×C17—42no—
37C22×C8×C17—21no—
38C17×C4⋊Q8—21no—
39C17×C4⋊Q8—21no—
40C17×C4⋊Q8—21no—
41C17×C4⋊Q8D412yes—
42C4×Q8×C17—21no—
43C4×Q8×C17—21no—
44C4×Q8×C17—21no—
45C4×Q8×C17—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17—21no—
48C2×C4⋊C4×C17—21no—
49C2×C4⋊C4×C17D411yescharacteristic
50C2×C42×C17—21no—
51C2×C42×C17C2311yescharacteristic
52#10587633—42no—
53#10587633—42no—
54C2×#63227—24no—
55C2×#63227—24no—
56C2×C4⋊Dic17D412yes—
57C2×C4×Dic17—21no—
58C22×#3622—21no—
59C22×#3622—21no—
60#10587560—21no—
61#10587560—21no—
62#10587560—21no—
63#10587560—21no—
64C17×C4⋊C8—21no—
65C17×C4⋊C8D412yes—
66C17×#103—24no—
67#10587502—21no—
68#10587480—28no—
69C4⋊#3622—21no—
70C4⋊#3622—21no—
71C4×#3622—21no—
72C4×#3622D412yes—
73C23×D17—22no—
74C2×Q8×C17—21no—
75C2×Q8×C17—21no—
76C2×Q8×C17—22no—
77C2×Q8×C17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17—21no—
80C2×Q8×C17—41no—
81C2×Q8×C17—41no—
82C2×Q8×C17—41no—
83C2×Q8×C17—41no—
84C2×Q8×C17—41no—
85C2×Q8×C17D812yes—
86C22×C4×C17—21no—
87C22×C4×C17—21no—
88C22×C4×C17—41no—
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C22×C4×C17C2×D411yescharacteristic
92C22×Dic17—22no—
93C2×D68—22no—
94C2×D68—22no—
95C2×D68—44no—
96C2×D68—41no—
97C2×D68—41no—
98C2×D68—41no—
99C2×D68—41no—
100C2×D68—41no—
101C2×D68—42no—
102C2×D68—41no—
103C2×D68—41no—
104C2×D68—41no—
105C2×D68—42no—
106Q16×C17—48no—
107C2×C8×C17—22no—
108C2×C8×C17—41no—
109C2×C8×C17—41no—
110C4⋊C4×C17—22no—
111C4⋊C4×C17—21no—
112C4⋊C4×C17—22no—
113C4⋊C4×C17—21no—
114C4⋊C4×C17—41no—
115C4⋊C4×C17—41no—
116C4⋊C4×C17—41no—
117C4⋊C4×C17—41no—
118C42×C17—41no—
119C42×C17—41no—
120C42×C17C2×D411yescharacteristic
121C42×C17C2×D411yescharacteristic
122C42×C17C2×D411yescharacteristic
123C42×C17#4111yescharacteristic
124#63227—48no—
125#63227—84no—
126C4⋊Dic17—22no—
127C4⋊Dic17—22no—
128C4×Dic17—41no—
129C4×Dic17—41no—
130C2×#3622—22no—
131C2×#3622—42no—
132C2×#3622—41no—
133C2×#3622—42no—
134#10063—171nomaximal, Sylow
135C23×C17#12011yescharacteristic
136C22×D17—42no—
137C22×D17—42no—
138C22×D17—42no—
139C22×D17—44no—
140C22×D17—44no—
141Q8×C17—22no—
142Q8×C17—22no—
143Q8×C17—41no—
144Q8×C17—41no—
145Q8×C17—42no—
146Q8×C17—42no—
147Q8×C17—81no—
148Q8×C17—81no—
149C2×C4×C17—22no—
150C2×C4×C17—22no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—42no—
155C2×C4×C17—42no—
156C2×C4×C17—81no—
157C2×C4×C17—81no—
158C2×C4×C17#12011yescharacteristic
159C2×C4×C17#12011yescharacteristic
160C2×C4×C17#12111yescharacteristic
161C2×C4×C17#12111yescharacteristic
162C2×C4×C17#12111yescharacteristic
163C2×C4×C17#12711yescharacteristic
164C2×C4×C17C2×D811yescharacteristic
165C2×C4×C17#13512yes—
166C2×C4×C17#13511yescharacteristic
167C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
168C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C17#13712yes—
170C2×Dic17—42no—
171C2×Dic17—42no—
172C2×Dic17—42no—
173D68—44no—
174D68—82no—
175D68—81no—
176D68—81no—
177D68—82no—
178D68—81no—
179D68—81no—
180D68—81no—
181D68—81no—
182C8×C17—42no—
183#3622—42no—
184#3622—81no—
185C2×#486—171no—
186C2×#462D1711yescharacteristic, complemented
187#2548—171no—
188#1988—171no—
189C2×#331—171no—
190C2×Q8⋊C8—171no—
191#1279—171no—
192C22×C17#39211yescharacteristic
193C22×C17#44711yescharacteristic
194C22×C17#44911yescharacteristic
195C22×C17#45212yes—
196C22×C17#45512yes—
197C2×D17—84no—
198C2×D17—84no—
199C2×D17—82no—
200C2×D17—82no—
201C2×D17—82no—
202C4×C17—21no—
203C4×C17—21no—
204C4×C17—42no—
205C4×C17—82no—
206C4×C17—81no—
207C4×C17#44711yescharacteristic
208C4×C17#44911yescharacteristic
209C4×C17#46412yes—
210C4×C17#46611yescharacteristic
211C4×C17#46911yescharacteristic
212C4×C17#49511yescharacteristic
213C4×C17#49711yescharacteristic
214Dic17—82no—
215C22×Q16—21no—
216C2×C4⋊Q8C2×D1711yescharacteristic
217C2×#127—171no—
218C2×#124—341no—
219C2×C4×Q8C2×D1711yescharacteristic
220#486—342no—
221#486—342no—
222#462—22no—
223#462—22no—
224C2×C4⋊C8—171no—
225C2×C4⋊C8C2×D1711yescharacteristic
226C2×#103—21no—
227C2×#102—342no—
228C2×C4×C8—171no—
229#394—342no—
230#390—342no—
231#390—341no—
232C4×C4⋊C4—171no—
233#340—342no—
234#331—342no—
235Q8⋊C8—342no—
236C2×C17#164012yes—
237C2×C17#164112yes—
238C2×C17#199911yescharacteristic
239C2×C17#200411yescharacteristic
240C2×C17#201511yescharacteristic
241D17—162no—
242C22×Q8—21no—
243C22×Q8C22⋊D1711yescharacteristic
244C22×D4—172no—
245C22×D4—341no—
246C22×D4—341no—
247C2×Q16—24no—
248C2×Q16—42no—
249C2×Q16—42no—
250C22×C8—21no—
251C22×C8—341no—
252C22×C8—341no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—21no—
256C4⋊Q8C22⋊D1712yes—
257#127—341no—
258#127—341no—
259#127—341no—
260#127—341no—
261#124—682no—
262#124—682no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—172no—
271C2×C4⋊C4C22⋊D1711yescharacteristic
272C2×C22⋊C4—344no—
273C2×C22⋊C4—344no—
274C2×C42—21no—
275C2×C42—341no—
276C2×C42C22×D1711yescharacteristic
277C4⋊C8—21no—
278C4⋊C8—341no—
279C4⋊C8—341no—
280C4⋊C8C22⋊D1712yes—
281#103—24no—
282#102—348no—
283C4×C8—172no—
284C4×C8—341no—
285#95—341no—
286C17#1006311yescharacteristic, complemented, Sylow
287C24—342no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8#6322812yes—
300C2×D4—342no—
301C2×D4—342no—
302C2×D4—684no—
303C2×D4—682no—
304C2×D4—682no—
305C2×D4—681no—
306C2×D4—681no—
307C2×D4—681no—
308C2×D4—681no—
309C2×D4—681no—
310C2×D4—681no—
311C2×D4—681no—
312C2×D4—681no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—41no—
316C22×C4—342no—
317C22×C4D4×D1711yescharacteristic
318C22×C4D4×D1711yescharacteristic
319C22×C4C2×C22⋊D1711yescharacteristic
320Q16—48no—
321C2×C8—22no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—342no—
325C2×C8—682no—
326C2×C8—682no—
327C2×C8—681no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—342no—
337C4⋊C4—342no—
338C22⋊C4—688no—
339C22⋊C4—1364no—
340C42—41no—
341C42—41no—
342C42—681no—
343C42—681no—
344C42D4×D1711yescharacteristic
345C42#6325411yescharacteristic
346C42C2×C22⋊D1711yescharacteristic
347C42C2×C22⋊D1711yescharacteristic
348C23—682no—
349C23—684no—
350C23—682no—
351C23—684no—
352C23—682no—
353C23#1058762311yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358Q8—42no—
359Q8—42no—
360Q8—81no—
361Q8—81no—
362D4—684no—
363D4—1362no—
364D4—1361no—
365D4—1361no—
366D4—1362no—
367D4—1361no—
368D4—1361no—
369D4—1361no—
370D4—1361no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—682no—
381C2×C4—682no—
382C2×C4—682no—
383C2×C4#1058760412yes—
384C2×C4#1058760512yes—
385C2×C4C2×#6322811yescharacteristic
386C2×C4#1058761811yescharacteristic
387C2×C4#1058762311yescharacteristic
388C2×C4#1058762411yescharacteristic
389C2×C4#1058762511yescharacteristic
390C2×C4#1058762511yescharacteristic
391C2×C4#1058762611yescharacteristic
392C2×C4#1058763511yescharacteristic
393C2×C4#1058763611yescharacteristic
394C2×C4#1058763911yescharacteristic
395C8—42no—
396C8—682no—
397C8—1361no—
398C22—1362no—
399C22—1362no—
400C22—1364no—
401C22—1364no—
402C22—1362no—
403C22#1176271711yescharacteristic
404C22#1176294211yescharacteristic
405C22#1176295211yescharacteristic
406C22#1176297012yes—
407C22#1176298512yes—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—81no—
412C4—82no—
413C4—1362no—
414C4#1176294311yescharacteristic
415C4#1176295411yescharacteristic
416C4#1176303312yes—
417C4#1176304311yescharacteristic
418C4#1176306011yescharacteristic
419C4#1176319411yescharacteristic
420C4#1176320211yescharacteristic
421C2—2722no—
422C2#1452796812yes—
423C2#1452797512yes—
424C2#1452978011yescharacteristic
425C2#1452981011yescharacteristic
426C2#1452987511yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×#462C2
socleC23×C17#120
prime radicalC2×#10587560C22
hypercenterC2×#462D17
hypocenterC17#10063
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #14529737C2semidirect
5. C2×Q8⋊#3622C2semidirect
6. #14526623C2semidirect
7. C2×C17×#462C2semidirect
186. C2×#462D17semidirect
286. C17#10063semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#117631572
#145337832
#145297372
C2×#117625522
C2×Q8⋊#36222
#145266232
C2×C17×#4622
#1006317